Self-reference

Ever since Epimenides the Cretan (7th century B.C.) declared that All Cretans are liars, the notion
of self-reference became synonymous (and often erroneously) with that of paradox. However, most ubiquitous
self-reference which happens each time somebody says "I", is harmless. Following are a few gems from
Metamagical Themas.

Thit sentence is not self-referential because "thit" is not a word.

If this sentence didn't exist, somebody would have invented it.

This is not a complete. Sentence. This either.

This sentence will end before you can say "Jack Rob

Does this sentence remind you of Agatha Cristie?

Raymond Smullyan lists what he calls self-annihilating sentences
from a collection by Saul Gorn "S. Gorn's Compendium of Rarely Used Cliches." :

In a Spectator competition the following won a prize; subject: what would you most like to read on opening the morning paper?

OUR SECOND COMPETITION

The First Prize in the second of this year's competitions goes to
Mr Arthur Robinson, whose witty entry was easily the best of those
we received. His choice of what he would like to read on opening his
paper was headed, 'Our Second Competition', and was as follows: "The
First Prize in the second of this year's competitions goes to Mr Arthur
Robinson, whose witty entry was easily the best of those we received. His
choice of what he would like to read on opening his paper was headed
'Our Second Competition', but owing to paper restrictions we cannot
print all of it."

Here are some self-referential statements that I came across recently.

Break every rule.

All generalizations are misleading.

If somebody loves you, love them back unconditionally.

Computers are like lynxes in the sense that I cannot think of a
suitable analogy for either of them right now.

(The last one is by Scott Adams in one of his Dilbert books.)

Robert Arthur quoted from Desmond MacHale's "Comic
Sections". Said he, one of the cleverest things I have ever seen in a book went something like this. Page 347 was devoted to ERRATA and consisted of the following

Page 347: For ERRATA read ERRATUM

John Allen Paulos opens his book Mathematics and Humor with a paragraph that contains a story of a friend of his who took a speed-reading course. The friend noted this in a letter to his mother. His mother responded with a long, chatty letter in the middle of which she wrote, "Now that you've taken that speed-reading course, you've probably already finished reading this letter."

William Dunham mentions Stephen Bock's Description of a sheltered man and his dreams: "Reading was something Jay knew about only from books, yet he was quite anxious to experience it for himself."

A superb site IMPOSSIBLE OBJECTS (which I no longer find accessible) defined Impossible objects as

An impossible object is a collection of reasonable parts, put together in an impossible way.

Now think of it. Is it possible to put anything together in an impossible way?

At the end of World War II, (Stefan) Bergman found himself in France without any papers. This was always tricky, but at that time it was a matter of life and death: without papers, one could not obtain a rations card. Characteristically, Bergman found a mathematical solution to the problem. He went to the mayor of a small town outside Paris and convinced him to give Bergman a piece of paper saying, "This is to certify that Mr. Stefan Bergman has no papers." This of course was a paper, and with it Bergman was able to obtain his rations card.

Camiel Feij from the Netherlands sent me a delightful philosophical observation with a probable attribution to Isaac Bashevis Singer:

We have to believe in free-will, we have no choice.

An insert on self-reference appeared in Mathematics Magazine Vol 78, No 1, February 2005:

A Publishing Paradox

A recent Birkhäuser-Verlag book list included Unpublished Philosophical Essays/Kurt Gödel, edited by Francisco A. Rodrigez-Consuegra, 1995. If the title is accurate, it might make appropriate barbershop reading in that town where the barber shaves everyone who doesn't shave himself.

A quick search of Amazon.com turns up a wealth of similar material -- unpublished recordings of Elizabeth Schwartzkopf and Marian Anderson, unpublished letters from General Robert E. Lee to Jefferson Davis, and unpublished opinions of Warren Court. Author Michael McMullen is more scrupulously logical: The title of his book is The Blessing of God: Previously Unpublished Sermons of Jonathan Edwards.

But now come to think of it: when else but previously a book might or might not have been published? Is this specification not redundant?

Additional examples can be easily found on the Web. I came across the following at the David's Profession Jokes web site:

There are three kinds of mathematicians:
those who can count and those who can't.

There are two groups of people in the world;
those who believe that the world can be
divided into two groups of people,
and those who don't.

"I'm getting so absent-minded and forgetful. Sometimes in the middle of a sentence, I . . . "

In the September 2005 issue of Math Horizons the readers were asked to submit questions that contained their own answers. The results of the competition have been published in the April 2006 issue. Following are a few samples:

In 1978, Raymond Smullyan wrote a book about logical puzzles. What is the name of this book?
(Roger Nelsen)

I am the square root of -1. Who am I?

What would be the value of 190 in hexadecimal be?
(Head-Royce School Math Club)

Twenty-nine is a prime example of what kind of number?

The reciprocal of sqrt(2) is half of what number?

How many consonants are in one? How many consonants are in two? How many consonants are in three? (One can continue: how many letters are in four?)

What do you do to the length of an edge of a cube to find its volume?
(Rheta Rubenstein)

As reported by BBC, US President George W Bush was twice the man at the White House Correspondents' Association annual dinner on Saturday, 29 April, 2006. Playing up to his public persona, Mr. Bush sent himself up alongside impersonator, Steve Bridges. At some point Mr. Bridges volunteered for the president:

"Some of my critics in the international community call me arrogant. I will not even honour that with a response. Screw them!"

Small girl: Because, if I did, I should have to eat it - and I can't bear it!

A quote from "Conversations on Game Theory" by Haim Shapira (I translate from Hebrew):

Most things are more complex than it appears to you even if it appears to you that you understand this sentence.

Now returning to the origin of all the troubles, Epimenides the Cretan, you have probably
heard of the island where each person is either a Knight (and speaks only truth) or a Knave
(and hence always lies). Does it follow that Epimenides could not possibly live in such a place?

Four. The sentence contains three spelling mistakes, plus the false claim that it only contains
one mistake, making a total of four mistakes.

The second question, paradoxically, cannot be answered. It contains only two spelling mistakes but
claims to contain three mistakes; therefore that claim is wrong and it actually contains three
mistakes - except that if it contains three mistakes then the claim that it contains three mistakes
is correct, and so it only contains the two spelling mistakes, in which case...!

Regarding your answer in the Self-reference section I believe that you are in error when you give the answer that the statement, "Their are three misteaks to this sentence", is unanswerable.

The three mistakes are:

Spelling error; Their for there.

Spelling error; Misteak for mistake.

Grammatical error; Use of Their for there.

I appreciate the work that has been done on all the sections of this web page. Keep up the work.

Richard Bissell

A response came from Italy:

hallo!

I just wanted to point out something about Richard Bissel remark to your answer concerning the sentence "Their are three misteaks in this sentence".

He says that error three is: "3. Grammar error, use of their for there."

I think that if we accept this as an error then the word 'their' is no longer mispelled and therefore error one isn't really an error.

Thanks again for your great site.

Rita

Here's a reaction to Rita's letter:

Nice site.

I was looking at your page on self-reference and I came upon the David Wells questions and the additional readers' comments.

First, Richard Bissell did not say that all three mistakes coexisted, so Rita's comment is not a valid criticism.

Mine, however, is. There are no mistakes in either sentence. While they contain incorrect spellings, falsehoods, and a violation of a grammatical rule, none of these were accidental, and therefore they cannot legitimately be considered mistakes.